Let W be the set of all vectors of the form shown, where a,...

Let $W$ be the set of all vectors of the form shown, where $a, b,$ and $c$ represent arbitrary real numbers. In each case, either find a set $S$ of vectors that spans $W$ or give an example to show that $W$ is not a vector space. $\left[\begin{array}{l}{-a+1} \\ {a-6 b} \\ {2 b+a}\end{array}\right]$

Let $W$ be the set of all vectors of the form shown, where $a, b,$ and $c$ represent arbitrary real numbers. In each case, either find a set $S$ of vectors that spans $W$ or give an example to show that $W$ is not a vector space.
$\left[\begin{array}{l}{-a+1} \\ {a-6 b} \\ {2 b+a}\end{array}\right]$

Key Concepts

Linear Combination

A linear combination involves taking scalar multiples of vectors and adding them together. This concept is fundamental for expressing arbitrary vectors in a space in terms of a given set of vectors, and for determining whether a given set of vectors spans a space.

Affine Space

An affine space is similar to a vector space, but it does not necessarily include the zero vector because it can be seen as a translation of a vector subspace. When a set of vectors is defined with a constant term or offset (resulting in non-zero vectors for the 'zero' parameters), the set might form an affine space rather than a true vector subspace.

Spanning Set

A spanning set for a vector space or a subset of a vector space is a collection of vectors such that every vector in the set can be written as a linear combination of those vectors. Identifying a spanning set is central to understanding the structure and dimensionality of a space or subspace.

Vector Space

A vector space is a set together with two operations—vector addition and scalar multiplication—that satisfy a list of axioms, such as the existence of a zero vector, associativity, and distributivity. Verifying that a set is a vector space involves checking that it contains the zero vector and is closed under both addition and scalar multiplication and that it satisfies the other vector space axioms.

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